Center manifold theory for the motions of camphor boats with delta function (Q2025667)

In this article, based on the results in [\textit{M. Nagayama} et al., Physica D 194, No. 3--4, 151--165 (2004; Zbl 1076.76608); \textit{N. J. Suematsu} et al., Phys. Rev. E 81, No. 5, Article ID 056210, 5 p. (2010; \url{doi:10.1103/PhysRevE.81.056210})], the problem of self-driven movement of a camphor boat is considered. Camphor boats are self-driving particles that interact with each other by changing the surface tension of the water surface by camphor molecules. The mathematical model is described by a system of differential equations. The first equation is a Newton's equation of motion of a camphor boat with a surface tension of water; the second is the reaction-diffusion equation containing the delta function. The main result of the work complements the research in [\textit{S. I. Ei} et al., Physica D 165, No. 3--4, 176--198 (2002; Zbl 1098.35542)]. The authors consider the interaction between two pulses with very small velocity near the bifurcation point in the reaction-diffusion system. Each impulse can be approximated by a stationary solution traveling wave type. A new approach to the center manifold in the (\(H_1\))*-framework has been developed to overcome the mathematical difficulties in the reduction the reaction-diffusion system of equations with a spatial discontinuity in the model (with the Dirac delta function) to the system of ordinary differential equations. Numerical simulations are performed.